Observations of exotic structures in the J / ψ p channel, which we refer to as charmonium-pentaquark states, in Λ 0 b → J / ψ K − p decays are presented. The data sample corresponds to an integrated luminosity of 3     fb − 1 acquired with the LHCb detector from 7 and 8 TeV p p collisions. An amplitude analysis of the three-body final state reproduces the two-body mass and angular distributions. To obtain a satisfactory fit of the structures seen in the J / ψ p mass spectrum, it is necessary to include two Breit-Wigner amplitudes that each describe a resonant state. The significance of each of these resonances is more than 9 standard deviations. One has a mass of 4380 ± 8 ± 29     MeV and a width of 205 ± 18 ± 86     MeV , while the second is narrower, with a mass of 4449.8 ± 1.7 ± 2.5     MeV and a width of 39 ± 5 ± 19     MeV . The preferred J P assignments are of opposite parity, with one state having spin 3 / 2 and the other 5 / 2 .